Divisibility Rules

Please note that the divisibility rule 7 is tentative.   There is technically no shortcut.

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Divisibility rules are shorthand ways of division to tell if one number is divisible or not.  They help tell whether the specific number you are looking for is prime or not. The many divisibililty rules help many mathematicians and geniuses determine prime numbers, even if the number is beyond big.

Let's look at some divisibility rules:

1-20

First 10 divisibility rules

Divisor	Test	Example
1	Every number is divisible by one.	4623 is divisible by one, and even the greatest prime integer is greater than one.
2	Every number apart from 0 which ends with 0, 2, 4, 6 or 8 is divisible by two.	65156151594 is divisible by two because the units digit is 4, and 1597534568852 is divisible by two because the units digit is 2.
3	A number is divisible by 3 if the sum of its digits is divisible by 3.	Is 12423 divisible by 3? Add all the digits together, we get 1 + 2 + 4 + 2 + 3 = 12. 12 is divisible by 3, so 12423 is, too.
4	A number is divisible by four if and only if its last two digits fo the number are divisible by four.	156128 is divisible by four, and 6416 is divisible by four. The last two digits are 28 and 16, respectively, both of which are divisible by 4.
5	A number is divisible by 5 if and only if the last digit is a 0 or 5.	213478765 is divisible by 5 because the last digit is 5.
6	Because its prime factorization is 2 x 3, it should be divisible by those two numbers.	Is 4,044 divisible by 6? 4,044 is even, so it is divisible by 2. 4 + 4 + 4 is 12, so it is divisible by 3. Therefore, 4,044 is indeed a multiple of 6.
7 (method 1)	The first method is to take the last digit of the number, multiply it by two, then subtract it to the remaining digits of the number. Repeat the process until the result can be easily identified.	Is 3409 is divisible by 7? Take 9 from 3409. Multiply 9 by 2 (9*2=18) Subtract the doubled digit to the remaining number (340-18=322) Repeat the progress, this time take 2 from 322. Multiply 2 by 2 (2*2=4) Subtract the doubled digit to the remaining number (32-4=28) 28 is divisible by 7, so 3409 is divisible by 7. (The real answer is 487)
7 (method 2)	Sort the numbers in blocks of three. Then, add the first group from the right, subtract the second group, then add the third, subtract the fourth, and so on. This is called alternating sum. If the result number is the multiple of 7, then the number is divisible by 7.	Is 1702906247 divisible by 7? Group the numbers into blocks of three: 247, 906, 702, 1 Form the alternating sum: 247 - 906 + 702 - 1 = 42 Since 42 = 7 * 6, 42 is divisible by 7, thus 1702906247 is, too.
8	It is very simple, check whether the last three digits of the number is divisible by 8, if it is, then the number is divisible by 8.	Is the number 7,377,473,496 divisible by 8? Take the last three digits of the number. The last three digit is 496, so check whether it is divisible by 8. 496 is divisible by 8, So the number 7,377,473,496 is divisible by 8. Is the number 546,632,318 divisible by 8? The last three digit is 318. However, 318 is not divisible by 8, so the number 546632318 is not divisible by 8.
9	Add all the digits together. The only step different is to check if it is the multiple of 9 instead of 3.	Is 14625 divisible by 9? Add all the digits together: 1 + 4 + 6 + 2 + 5 = 18 18 is divisible by 9, so 14625 is, too.
10	If the last digit of the number is 0, it is divisible by 10.	158375209580 is divisible by 10 because the last digit is 0.

Beyond 10

Divisibility by 11

Check the number, if the difference of sum of digits at odd places and sum of digits at even places is 0 or divisible by 11, then the number is divisible by 11.

Example: Is 527901 divisible by 11?

1. First, check the number.
2. Add the digits at odd places (in this case, 5+7+0=12)
3. Add the digits at even places (in this case, 2+9+1=12)
4. Subtract the sum of digits at odd places and sum of digits at even places (12-12=0)
5. The result is 0, so 527901 is divisible by 11. (The real answer is 47991)

Divisibility by 12

Because its prime factorization is 2^2 x 3, it needs to be divisible by 4 and 3.

Example:

Is 9,180 divisible by 12?

1. 9,180 ends in 80 (4*20), so it is divisible by 4.
2. 9 + 1 + 8 + 0 is 18, so it is divisible by 3.
3. Therefore, 9,180 is a multiple of 12.

Divisibility by 13

There is no shortcut for testing divisibility of this particular number, either.

Method 1

The first method is to take the last digit of the number, multiply it by four, then add it to the remaining digits of the numbers. Repeat the process until the result can be easily identified. This is similar to divisibility of 7.

Example:

Take 85969 for example.

1. Take out 9 from 85969.
2. Multiply if by four. (9 x 4 = 36)
3. Add it to the remaining digits. (8596 + 36 = 8632)
4. Take out 2 from 8632 and multiply it by four. (2 x 4 = 8)
5. Add it to the remaining digits. (863 + 8 = 871)
6. Take out 1 from 871 and multiply it by four. (1 x 4 = 4)
7. Add it to the remaining digits. (87 + 4 = 91)
8. 91 is divisible by 13, so 85969 is also divisible by 13.

Method 2

The second method, also similar to divisibility of 7, is use the alternating sums. After sorting the numbers in blocks of three, add the first group from the right, subtract the second group, then add the third, subtract the fourth, and so on. At last, check if the result is the multiple of 13 (If needed, use Method 1 to further reduce down a 3-digit number).

Example:

1. Take 3,692,513,279 for example, $-3 + 692 - 513 + 279 = 455$
2. After that, use Method 1 to determine if 455 is a multiple of 13.
3. Take out 5 from 455 and multiply it by 4. (5 x 4 = 20)
4. Add it to the remaining digits. (45 + 20 = 65)

65 is divisible by 13. Therefore, 3,692,513,279 is also divisible by 13.

Divisibility by 14

As 14 is the product of 2 and 7, perform divisibility checks of both 2 and 7.

Divisibiltiy by 15

Check if the number is divisible by 3 and 5.

Divisibility by 16

Same as divisibility by 8, except check if the last 4 digits are divisible by 16 rather than the last 3.

Divisibility by 18

Check if the number is divisible by 2 and 9.

Divisibility by 19

To check if a number is divisible by 19, take off the last digit and double it. Add it to the remaining digits. If that number is divisible by 19, so is the original number.

Example:

209. Take off the last digit (9) and double it (18). Add 18 to the remaining digits (20) to get 38. You can do the process again. Take off the last digit (8) and double it (16). Add 16 to the remaining digits (3) to get 19. 209 is divisible by 19.

Divisibility by 20

If the last digit of the number is 0 and the tens place is even, it's divisible by 20.

Divisibility by 21

Method 1

21 ends in 1, so you can take the last digit, multiply it by 2, and subtract it from the remaining digits of the number just like 7.

Method 2

Do the test for 3 and the test for 7. It the number is divisible by both, it's divisible by 21.

Divisibility by 29

To check if a number is divisible by 29, do the same as with 19 except when you take off the last digit, triple it rather than double it. This trick works with every number ending in 9. For 39 multiply it by 4, for 49 multiply it by 5, for 59 multiply it by 6 and so on.

Divisibility by 30

If the sum of digits is a multiple of 3, and the last digit of the number is 0, then it's divisible by 30.

Divisibility by 32

Do the same as with 16, but check that the last 5 digits are divisible by 3 rather than the last 4.

Divisibility by 39

Method 1

39 ends in 9, so you can do the test for 29 but this time multiply the last digit by 4 rather than 3.

Method 2

Do the test for 13 and the test for 3. If they both turn out positive, then the original number is divisible by 39.